An Approach to Estimation and Extrapolation with Possible Applications in an Incompletely Specified Environment

In problems arising in the analysis of systems operating in an unknown or random environment, probability theory and statistical decision theory usually have been employed to obtain solutions. Recently, however, some researchers have recognized some inadequacies and difficulties in the application of these theories. For this reason we seek alternative techniques for application to two frequently occurring problems which arise under uncertainty: the problems of “estimation” and “extrapolation.” In our approach, a probability-statistics model is not assumed. Instead, we formalize what is meant by a rational choice of an estimation or extrapolation procedure by listing a set of intuitively reasonable or desirable properties (axioms) which incorporate the definition of and our goals in performing estimation and extrapolation. The decision rules which satisfy these axioms are then determined; to get some idea of the behavior of these rules we study their performance on data from both statistical and incompletely specified sources. The estimation and extrapolation methods developed here might be useful in situations in which either a probabilistic model is not meaningful, or in cases in which too little data is available to infer a priori statistical knowledge.